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Eight Embodiment (Posted on 2014-05-25) Difficulty: 3 of 5
Determine the respective minimum values of a positive integer N for each p = 1 to 9 inclusively, such that:
  • N is composite, and:
  • All positive divisors of N (including itself but, with the exclusion of 1) contain the digit p.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (2 votes)

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Solution computer solution | Comment 4 of 6 |

DefDbl A-Z
Dim fct(20, 1), f, factr, dgt$
Function mform$(x, t$)
  a$ = Format$(x, t$)
  If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
  mform$ = a$
End Function

Private Sub Form_Load()
 Form1.Visible = True
 Text1.Text = ""
 For dg = 0 To 9
   dgt$ = LTrim(Str(dg))
   For n = 2 To 1000000
     factr = 1
     f = factor(n)
     If f > 1 Or fct(1, 1) > 1 Then
      If f > 2 Or f = 2 And (fct(1, 1) > 1 Or fct(2, 1)) > 1 Then
      If isitgood(1) Then
       Text1.Text = Text1.Text & mform(dg, "#0") & mform(n, "#####0" & "    ")
       vlu = n
       Do
         pf = prmdiv(vlu): vlu = vlu / pf
         Text1.Text = Text1.Text & Str(pf)
       Loop Until vlu = 1
       Text1.Text = Text1.Text & Chr(13) & Chr(10)
       DoEvents
       Exit For
      End If
      End If
     End If
   Next
 Next
End Sub

Function isitgood(wh)
  bad = 0
  For usetimes = 0 To fct(wh, 1)
    factr = factr * fct(wh, 0) ^ usetimes
    If wh = f Then
     fs$ = LTrim(Str(Int(factr + 0.5)))
     If InStr(fs$, dgt$) = 0 And factr <> 1 Then bad = 1: Exit For
    Else
     If isitgood(wh + 1) = 0 Then bad = 1: Exit For
    End If
    factr = factr / fct(wh, 0) ^ usetimes
  Next
  If bad Then isitgood = 0 Else isitgood = 1
End Function

Function factor(num)
 diffCt = 0: good = 1
 n = Abs(num): If n > 0 Then limit = Sqr(n) Else limit = 0
 If limit <> Int(limit) Then limit = Int(limit + 1)
 dv = 2: GoSub DivideIt
 dv = 3: GoSub DivideIt
 dv = 5: GoSub DivideIt
 dv = 7
 Do Until dv > limit
   GoSub DivideIt: dv = dv + 4 '11
   GoSub DivideIt: dv = dv + 2 '13
   GoSub DivideIt: dv = dv + 4 '17
   GoSub DivideIt: dv = dv + 2 '19
   GoSub DivideIt: dv = dv + 4 '23
   GoSub DivideIt: dv = dv + 6 '29
   GoSub DivideIt: dv = dv + 2 '31
   GoSub DivideIt: dv = dv + 6 '37
   If INKEY$ = Chr$(27) Then s$ = Chr$(27): Exit Function
 Loop
 If n > 1 Then diffCt = diffCt + 1: fct(diffCt, 0) = n: fct(diffCt, 1) = 1
 factor = diffCt
 Exit Function

DivideIt:
 cnt = 0
 Do
  q = Int(n / dv)
  If q * dv = n And n > 0 Then
    n = q: cnt = cnt + 1: If n > 0 Then limit = Sqr(n) Else limit = 0
    If limit <> Int(limit) Then limit = Int(limit + 1)
   Else
    Exit Do
  End If
 Loop
 If cnt > 0 Then
   diffCt = diffCt + 1
   fct(diffCt, 0) = dv
   fct(diffCt, 1) = cnt
 End If
 Return
End Function

Function prmdiv(num)
 Dim n, dv, q
 If num = 1 Then prmdiv = 1: Exit Function
 n = Abs(num): If n > 0 Then limit = Sqr(n) Else limit = 0
 If limit <> Int(limit) Then limit = Int(limit + 1)
 dv = 2: GoSub DivideIt
 dv = 3: GoSub DivideIt
 dv = 5: GoSub DivideIt
 dv = 7
 Do Until dv > limit
   GoSub DivideIt: dv = dv + 4 '11
   GoSub DivideIt: dv = dv + 2 '13
   GoSub DivideIt: dv = dv + 4 '17
   GoSub DivideIt: dv = dv + 2 '19
   GoSub DivideIt: dv = dv + 4 '23
   GoSub DivideIt: dv = dv + 6 '29
   GoSub DivideIt: dv = dv + 2 '31
   GoSub DivideIt: dv = dv + 6 '37
 Loop
 If n > 1 Then prmdiv = n
 Exit Function

DivideIt:
 Do
  q = Int(n / dv)
  If q * dv = n And n > 0 Then
    prmdiv = dv: Exit Function
   Else
    Exit Do
  End If
 Loop

 Return
End Function

finds

 p    N      prime factors of N
 0 10201     101 101
 1   121     11 11
 2   254     2 127
 3    39     3 13
 4  1849     43 43
 5    25     5 5
 6 16043     61 263
 7   497     7 71
 8  6889     83 83
 9  1691     19 89
 
(I thought for completeness, we might include zero.) 
 
Seeking candidates for N with more than 3 divisors: 
(by using
If f > 2 Or f = 2 And (fct(1, 1) > 1 Or fct(2, 1)) > 1 Then
)
   1   1859     11 13 13
   2  32258     2 127 127
   3  34917     3 103 113
   5  75815     5 59 257
   7  53179     7 71 107
   9 150499     19 89 89
   
Again, only the prime factors are shown, but of course all the divisors may be found by multiplication, such as:

19*89=1691, 89^2=7921, in addition to 19, 89 and 150499 itself.

  Posted by Charlie on 2014-05-25 14:15:40
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